Consider a risk-free zero-coupon bond
Face value (final payoff) of \(F\) with a maturity of \(T\) years
If the interest rate for maturity \(T\) is \(r_{T}\), then the current price of this bond is:
\[ P_{0} = \frac{F}{(1+r_{T})^{T}} \]
Notation \(r_{T}\) implies variation in \(r\) over time
Yield curve plots the term structure:
Notation \(r_{T}\) implies variation in \(r\) over time
Yield curve plots the term structure:
\[ P_{0} = \sum_{t=1}^{T} \frac{C}{(1+r_{t})^{t}} + \frac{F}{(1+r_{T})^{T}}, \]
\[ P = \sum_{t=1}^{T} \frac{C}{(1+YTM)^{t}} + \frac{F}{(1+YTM)^{T}}, \]
\[ 92.64 = \sum_{t=1}^{10} \frac{5}{(1+0.06)^{t}} + \frac{100}{(1+0.06)^{10}}, \]
\[ 108.11 = \sum_{t=1}^{10} \frac{5}{(1+0.04)^{t}} + \frac{100}{(1+0.04)^{10}}, \]
Note, YTM collapses \(r_{1}\) through \(r_{T}\) to a singular discount rate
Suppose we want to solve for entire yield curve
If we want to know \(T\) interest rates over the yield curve, we need \(T\) bonds (roughly)
Once we have interest rates for entire yield curve, can price any risk free bond
Example:
\[ P_{1} = \frac{C_{1} + 100}{(1+r_{1})}\\ P_{2} = \frac{C_{2}}{(1+r_{1})} + \frac{C_{2} + 100}{(1+r_{2})^2}\\ P_{3} = \frac{C_{3} }{(1+r_{1})} +\frac{C_{3} }{(1+r_{1})^2} + \frac{C_{3} + 100}{(1+r_{3})^3} \]
Consider a contract in which we agree to lend money for 1 year, beginning 1 year in the future (repayment 2 years in the future)
Generally, we will call \(f_{j,k}\) the forward rate which applies from period \(j\) to \(k\)
A comment on notation: we use \(r_{3}\) to mean today’s prevailing interest rate for a 3-year horizon. This is different from BKM (in which \(r_{3}\) refers to the 1-year rate which will prevail three years from now)
\[ (1+r_{1})(1+f_{1,2}) = (1+r_{2})^{2} \]
\[ f_{1,2} = \frac{(1+r_{2})^{2}}{(1+r_{1})} - 1 \]
\[ (1+r_{T})^{T} = (1+r_{t})^{t}(1+f_{t,T})^{T-t}\\ f_{t,T} = \left[\frac{(1+r_{T})^{T}}{(1+r_{t})^{t}}\right]^{\frac{1}{T-t}} - 1 \]
For example, \(r_{1} = 5\%\), \(r_{2} = 6\%\):
\[ f_{1,2} = \left[\frac{(1.06)^{2}}{(1.05}\right] - 1 = 7.01\% \]
Suppose investors are risk-neutral (or the future is certain)
Then, forward rates should be equal to the expectations of interest rates (why? - what trade could you do if not?)
Hence, long-term rates will represent a geometric average of current and future expected short rates
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) = (1+r_{0,1})E(1+r_{1,2}) \]
Now what if the investor is risk-averse and investing over two periods
Faces a choice between a sure two period rate, or a roll-over of a short-term bond
Recall that a risk-averse investor prefers a sure thing if the expected returns are equal
In equilibrium, two period investors bid down the long-term rate:
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) < (1+r_{0,1})E(1+r_{1,2}) \]
More commonly, we consider risk-averse one-period investor (needs liquidity at t=1)
Face a similar choice: a sure one-period rate, or a two-period bond that they sell in period 1
Recall that a risk-averse investor prefers a sure thing if the expected returns are equal
In equilibrium, one-period investors bid down the short-term rate:
\[ (1+r_{2})^{2} = (1+r_{0,1})(1+f_{1,2}) > (1+r_{0,1})E(1+r_{1,2}) \]
Commonly,upward sloping yield curves are interpreted as evidence of:
Define the liquidity premium as the difference between the forward rate and the expected future short rate
\[ K = (1+ f_{t,t+1}) - E(1+r_{t,t+1}) \]
The larger this is, the more that you pay for “certainty” of next period’s short-rate
Can generate common yield curve patterns under different expectations and values of K
Need a single measure that tells us the interest rate sensitivity of a bond/portfolio
Duration is a measure of the effective maturity
The weighted average of the amount of time until each payment is received, with the weights proportional to the present value of the payment
Duration is shorter than maturity for all bonds except zero coupon bonds
Duration is equal to maturity for zero coupon bonds
\[ D = \sum_{t=1}^{T} t \times w_{t} \\ w_{t} = \left[\frac{CF_{t}}{(1+YTM)^{t}}\right] / P, \]
where YTM is the yield to maturity, \(P\) is the price, and \(CF_{t}\) is the cashflows at time \(t\)
\[ \frac{\Delta P}{P} = -D\frac{\Delta (1+YTM)}{(1+YTM)}. \]
\[ \frac{\Delta P}{P} = -D^{*} \Delta YTM \]
Often, duration is important for investors who want to “immunize”" themselves from interest rate exposure
Consider our Florida Pension Fund case
Suppose FL would like to ensure it is hedged against interest rate changes
Invest in a bond portfolio with a duration equal to that of its liabilities
A couple of possibilities:
What are some issues with those strategies?
Duration is a local approximation
Duration measures sensitivity to changes in yield to maturity
Consider the case of Orange County, 1994
Bob Citron managed county funds as a large bet on interest rates
Borrowed large amounts in short term markets and reinvested in 5 year notes
Via leverage, portfolio had an effective duration of 7.4 years (2.74 duration of assets \(\times\) 2.7 leverage)
Four consecutive rate hikes by the FOMC
The OC has a $1.6 billion loss on a $7.5 billion portfolio (before leverage), declares bankruptcy
Question: How does this 1.6BN loss compare to expectations?
A useful application of VaR and duration
Fixed v. floating interest rate swap
One party pays a floating interest rate X notional value
Other party pays a fixed interest rate (the swap rate) X notional value
Exchanges made periodically over a fixed term
Fixed interest rate is set so that no cash exchanges hands at initiation
Swaps provide synthetic immunization
Note, counterparty which pays fixed and receives floating is short (bonds)
Consider the following three period fixed vs. floating rate swap
Lending or borrowing at forward rates can be achieved synthetically, or in futures market
Consider cash flows from a hedged counterparty who received fixed and pays floating
C is set to make the net present value of these streams equal to zero
This is known as the swap rate and is a weighted average of the spot and forward rates
\[ 0 = \frac{C- r_{0,1}}{(1+r_{1})} + \frac{C- f_{1,2}}{(1+r_{2})^{2}} + \frac{C- r_{2,3}}{(1+r_{3})^{3}} \]